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In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a singular matrix where the determinant is 0. In this process, information is lost and cannot be regained. For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive definite , which means that a ⋅ a {\displaystyle \mathbf {a} \cdot \mathbf {a} } is never negative, and is zero if and only if a = 0 {\displaystyle \mathbf {a} =\mathbf {0} } , the zero vector.
However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again ...
In semigroup theory, the absorbing element is called a zero element [1] [2] because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.
Properties (Dl) and (Dr) express biadditivity of φ, which may be regarded as distributivity of φ over addition. Property (A) resembles some associative property of φ. Every ring R is an R-bimodule. So the ring multiplication (r, r′) ↦ r ⋅ r′ in R is an R-balanced product R × R → R.
Former NHL player Paul Bissonnette is fine after getting into an altercation with six men at a local Scottsdale restaurant. (Photo by Cliff Hawkins/Getty Images for The Match) (Cliff Hawkins via ...
Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers C. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor.
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